Lecture 6 Events and the Field of Events The structure of this course follows a natural progression. We begin with the concept of a random variable, since it is an action that results in numbers. If the action is sufficiently known, then one can identify the collection of numbers that may possibly result prior to even taking the action. If the action is not sufficiently known, then it may or may not be able to infer properties of this collection from data that has been obtained by performing the action. Example 1 Suppose that the action that will be taken is to measure a person’s EEG for a period of 500 seconds at a sampling rate of 20 samples per second. Hence, this action corresponds to a random variable whose dimension is (20 samples/second)(500 seconds) = 10,000 samples (or numbers). Suppose further, that we know that any number cannot exceed 5 ± volts, since the amplifier is limited to that range. Finally, suppose that we have 32-bit resolution. At this fine level of resolution we can assume that essential any number in the interval ] 5 , 5 [-volts is possible. And so, without even performing the measurement of EEG, we know that the sample space for random variable X = ) , , , ( 000 , 10 2 1 X X X  is S X = } ] 5 , 5 [ | ) , , , {( 000 , 10 2 1 k every for x x x x k-∈  . Now, suppose that the range of values is not known, and for a given person we have the EEG measurement shown in the top plot of Figure 1below.0 50 100 150 200 250 300 350 400 450 500-0.50 0.5 EEG EEG Ch.47 Figure 1. The plot shows a 500-second measurement of EEG recorder from electrode #47. The sample rate is 20 samples/second. From this plot, one might be tempted to assume that the sample space for any element of X is ] 5 .0 , 5 .0 [-volts. Note, however, that this is only one sample of X . Consequently, it might not capture the entire range of values that any X k can take on. At this point one should either consult with the person who took the measurements to verify that, indeed, the permissible range is ] 5 .0 , 5 .0 [-volts, or presume that the range is larger. If one has no idea how large it might be, then the safest assumption would be ) , ([ ∞-∞ . □ Example 2 Suppose that you have assigned a problem to 10 students in the class, and that the answer is either correct or wrong (i.e. there is no partial credit. The act of recording the scores of the 10 students is X = ) , , , ( 10 2 1 X X X  is S X = } } 1 ,0 { | ) , , , {( 10 2 1 k every for x x x x k ∈  . Now, suppose that you are to report the average score to the principal. The act of computing this score is X X k k ∆ = = ∑ 10 1 10 1 . [Note: the symbol ∆ = ia a defined equality; that is, the symbol X is defined by the operation associated with it- in this case, the averaging operation.] Q: What is X S ? A: X S = ______________________________________________________

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